XPost: sci.math   
   From: invlaid@invalid.com   
      
   On 1/15/2017 8:11 AM, Anton Shepelev wrote:   
   > I wrote:   
   >   
   >> Is there a symbolic method to calculate the inten-   
   >> sity distribution on a flat  screen  from  a  flat   
   >> wavefront  after it reflected from a curved mirror   
   >> whose surface is described by an equation?   
   >>   
   >> The normal of the wavefront, and the relative  po-   
   >> sition of screen and mirror are known.   
   >   
   > Thank you for replies, Phil and Yuri.   
   >   
   > I  am  currently working on two analytical solutions   
   > under the assumptions of geomtrical optics (indepen-   
   > dent linear rays) of the simple case when the mirror   
   > deformations are so small that  not  only  secondary   
   > reflexions are impossible but also no concave region   
   > may focus parallel rays before they hit the screen.   
   >   
   > One version involves the caluclation of two orthogo-   
   > mal curvature radii, of which one is in the plane of   
   > the ray, and the other of only the second derivative   
   > of the mirror surface.  In the two-dimensional case,   
   > when the screen is placed  orthgonally  to  the  re-   
   > flected  rays  (for under the said assumptions their   
   > directions will not differ much), the  second  solu-   
   > tion gives the following intensity on the one-dimen-   
   > sional screen:   
   >   
   >    I(x) = I0 [ 1 - 2d Sec(alpha) S''(x Sec(alpha)) ]^-1   
   >   
   > where I0 is the intensity of the incoming  light,  d   
   > the distance to the screen, alpha the angle of inci-   
   > dence, and y = S(x) the one-dimensional mirror  sur-   
   > face.   
   >   
   > The  concave  parable, for example, will focus light   
   > uniformly, i.e. increase intensity  in  all  points.   
   > Does the resemble the truth?   
   >   
      
   yes that is true.   
      
   I was designing essentially an unfocused parabala that would guide light   
   toward a centrial shperical surface, with a cuttoff of +_45 degrees, and   
   the light was from any angle.  Goal was to have highest light captured   
   overall. (Non-imaging optics).   
      
   What I did one time was a 2 dimentional analysis, where I could specify   
   all the angles, angle of arival, surface angle, angle of reflection,   
   and I had a surface of intercept (photomultipler tube surface). so I   
   came up with an equation relating incoming ray angle and offset, to   
   intercept on photomultiplier tube surface.   
      
   With that I could come up with equations that would determine if the ray   
   (from x offset at angle k) would hit the photomultiplier surface.   
   The variable was the curvature of the surface, from squared to quartic.   
   I would change that run to run to see what type of surface curve works   
   best, turned out the 4th order worked better. (photomultiplier has light   
   bulb shaped surface)   
      
   There was a boundry set of acceptance of +- 45 degrees, so the mirror   
   would bounce the ray out if more than 45 degrees and not hit the   
   photomultiplier.   
      
   the math for such was stright forward, some complexity. I did not sum up   
   the intensity at points though, I guess that is another step where one   
   would sum up the # of rays at a location within an acceptance angle at   
   the surface.   
      
   that was all 2 dimentional. I did not go for exact for 3 dimentional, as   
   rotating the 2 dimentional solution was a first order approximation, and   
   the additional cases it missed did not seem to be that large.   
      
   I did double check this using a strip of reflective surface cut one of   
   those skylight light tunnels on top of a paper with the shapes of the   
   PMT, and cutoffs using a flashlight. Cheap and easy.   
      
   --- SoupGate-Win32 v1.05   
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